matrix product equals 0

Question

Be $A,X$ two matrices of the same order $n$ . Find the necessary and sufficient condition for $A$ so that there exists $X$ with the property $AX=XA=0_n$. I believe that rank A =1 is the condition but i can t really prove that it is sufficient. for obvious reasons, it is necessary.

Answer 1

Either $X$ or $A$ must be 0. You know this because anything multiplied by the "$0$" matrix is $0$. So, either can be a arbitrary matrix, and as long as one of them is $0$, the answer will be $0$.

Set $X=0$ because this will give you the trivial solution and you can see A is an arbitrary invertible matrix by the IMT (Invertible Matrix Theorem) which states:

"A matrix $A$ is invertible iff it has only the trivial solution $X=0$."

Therefore, by the IMT, rank $A=n$ from the statement (also from IMT):

"A matrix $A$ is invertible iff rank $(A) = n$."

and you know the matrix is invertible because it has only the trivial solution, and if one part of the IMT is true, they all hold for that matrix since the conditions are equivalent.

So the rank of the matrix $A=n$, not 1.